In 1985, Thurston [1]} proposed “the Finite Riemann Mapping Theorem", claiming that the maps induced from circle packings converge to the Riemann mapping. This conjecture initiated a flurry of research about the convergence of maps constructed from circle packings to the corresponding conformal maps. Rodin-Sullivan [2]} firstly confirmed the uniform convergence of the piecewise linear maps on a simply connected domain in the plane induced from the standard hexagonal circle packings to the Riemann mapping. A series of results including He [3]}, Doyle-He-Rodin [4]} and He-Schramm [5]} proved the convergence of the first and second derivatives of the induced maps from the circle packings extended by Möbius transformations to their smooth counterparts. Eventually, He-Schramm [6]} showed the \(C^{\infty }\)  convergence of hexagonal circle packings to the Riemann mapping. It also leads to the study of discrete analogs of concepts and theorems in complex analysis and Riemann surfaces. See [7]} for a comprehensive survey on this topic.
